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A variational approach to the Navier-Stokes equations
Nicola Gigli, Sunra Mosconi
To cite this version:
Nicola Gigli, Sunra Mosconi. A variational approach to the Navier-Stokes equations. Bulletin des Sciences Mathématiques, Elsevier, 2012, pp.1-21. �hal-00769387�
A variational approach to the Navier-Stokes equations
Nicola Gigli ∗ Sunra J. N. Mosconi † December 3, 2011
Abstract
We propose a time discretization of the Navier-Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial datum in H1, the scheme converges to strong solutions in some interval [0, T ) and, if the datum satisfies the classical smallness condition, it produces the smooth solution in [0, ∞).
1
Introduction
We consider the Navier-Stokes equations on the d−dimensional flat torus Td= Rd/Zd: ∂tut+ (ut· ∇)ut+ ∇pt = ∆ut, in [0, ∞) × Td, ∇ · ut = 0, in Td∀t, u0 = u, in Td, (1)
where the initial datum u is a given divergence free vector field, say smooth.
The purpose of this paper is to present a time-discretization argument, inspired by the theory of gradient flows, which allows to quickly reproduce several known results about (1). The idea is the following. Fix a parameter τ > 0, which we think as time step and, given u, define its flow map R × Td3 (t, x) 7→ Xu
t(x) ∈ Td as the only solution of ∂tXtu = u ◦ Xtu, X0u = Id. Now minimize v 7→ 1 2 Z Td |∇v|2dLd+kv ◦ X u τ − uk2L2 2τ ,
among all L2 and divergence free vector fields v. It is not hard to check (see Proposition 3.1) that the unique minimum uτ satisfies
uτ− u ◦ Xu −τ
τ + ∇p
τ = ∆uτ, (2)
∗
Universit`e de Nice nicola.gigli@unice.fr
†
where pτ is identified, up to additive constants, by ∆pτ = ∇ · u ◦ X u −τ τ .
We claim that (2) is a time discretization of (1). Indeed, the term
uτ − u ◦ Xu −τ τ = uτ◦ Xu τ − u τ ◦ X−τu ,
is the time discretization of the convective derivative
∂tut+ (ut· ∇)ut= ∂t(ut◦ Tt) ◦ Tt−1, where here [0, ∞) × Td 3 (t, x) 7→ T
t(x) ∈ Td is the flow map (or particle-trajectory map) associated to (ut), i.e.:
∂tTt = ut◦ Tt, T0 = Id. and the pressure term satisfies
∆pτ = ∇ · u ◦ X u −τ τ = ∇ · u ◦ X u −τ − u τ ,
which is a time discretization of
∆pt= ∇ · (ut· ∇)ut = ∇ · ∂tut+ (ut· ∇)ut, the latter being the formula identifying the pressure in (1).
The idea is then to repeat the minimization procedure with uτ in place of u, then with the new minimizer in place of uτ and so on. This argument produces a discrete solution (uτt) and our goal is to show that letting τ ↓ 0 the discrete solutions converge, in a sense to be specified, to certain solutions of (1).
We remark that a time discretization based on (2) is not entirely new in this setting: O. Pirroneau [7] used the same equation (without pointing out its variational structure) in the setting of numerical analysis to investigate the rate of convergence of the discrete solutions under the assumption that a smooth solution of (1) exists on some interval [0, T ].
The authors wishes to thank L. Ambrosio, Y. Brenier and C. De Lellis for fruitful conver-sations.
2
Notation and preliminaries
With Tdwe denote the d−dimensional flat torus Rd/Zd. A time dependent vector field [0, ∞) 3 (t, x) 7→ ut(x) ∈ Rdwill be typically denoted by (ut), while we write utfor the static vector field x 7→ ut(x). The subscript t will never stand for time derivative, which will be usually denoted by ∂t. When not specified, the integral symbol without further specification on the domain will stand for integration over Td(resp. [0, +∞)) when performed w.r.t. the measure dLd(resp. dt). We will also shorten Lp(Td, Rd) with Lp when the space is clear from the context.
Given a smooth vector vector field u ∈ C∞(Td, Rd), the flow map R × Td3 (t, x) 7→ Xu t(x) ∈ Td is the unique solution of
∂tXtu(x) = u ◦ Xtu(x), X0u(x) = x.
The classic Cauchy-Lipschitz theory ensures that Xu is C∞ in space and time as soon as u is C∞.
Recall that for a Borel map T : Td → Td and a non negative Borel measure µ on Td, the Borel measure T#µ on Td is defined by T#µ(E) := µ(T−1(E)) for all Borel sets E ⊂ Td. We will frequently use the fact that ∇ · u = 0 implies (Xtu)#Ld= Ld for any t ∈ R.
Given a vector field u ∈ C∞(Td, Rd), its Helmholtz decomposition is given by u = ∇p + w,
where ∇ · w = 0 and R pdLd = 0. It is not hard to see that the Helmholtz decomposition is unique and is an orthogonal decomposition of L2. For the existence, just solve
∆p = ∇ · u,
by also ensuringR pdLd= 0, and define
w := u − ∇p.
Classical elliptic regularity theory ensures that p, w are C∞ as soon as u is.
Given u ∈ C∞(Td, Rd), with |∇u| we will always mean the Hilbert-Smith norm of its gradient, given by
|∇u|2(x) :=X i,j
|∂iuj(x)|2.
We can, and will, suppose that at any time t ≥ 0, the velocity field has zero mean value. Indeed, integrating over Td equation (1) and integrating by parts gives for any solution of (1) it holds
d dt
Z
utdLd= 0 ∀t ≥ 0.
Thus, if v := R udLd, one can look for solutions (w
t) of (1) with initial data w = u − v, thus having zero mean velocity for any t ≥ 0. Letting then ut(x) = wt(x − vt) + v, it is easily checked
that (ut) is a solution of the original problem. The additional condition R
TdudL
d = 0 implies the first of the two frequently used estimates
kukL2∗(Td)≤ C k∇ukL2(Td),
k∇ukL2∗(Td)≤ C k∆ukL2(Td),
(3)
where 2∗ = d−22d , while the second one follows from standard elliptic estimates, since ∇u has zero mean on Tdby periodicity.
To show the convergence of the discretization scheme, we will use the Aubin-Lions lemma: Lemma 2.1 (Aubin-Lions) Let X ⊂ Y ⊂ Z be three Banach spaces such that: X and Z are reflexive, the embedding of X in Y is compact and the one of Y into Z is continuous. Then for any p, q ∈ (1, ∞) the space
n
u ∈ Lp([0, T ], X) : ∂tu ∈ Lq([0, T ], Z) o
,
is compactly embedded in Lp([0, T ], Y ).
For a proof see for instance [10] Chapter 3, Theorem 2.1.
3
Discrete solutions
Let u ∈ C∞(Td, Rd) be a smooth vector field and τ > 0. The functional F (v; u, τ ) is defined as F (v; u, τ ) := Z Td |∇v|2dLd+1 τ Z Td |v ◦ Xu τ − u|2dLd. (4)
Proposition 3.1 Let u ∈ C∞(Td, Rd) be a smooth vector field such that ∇ · u = 0 and τ > 0. Then there exists a unique minimizer uτ of v 7→ F (v; u, τ ) in the class of L2 vector fields such that ∇ · v = 0. The minimum uτ is C∞ and satisfies:
uτ− u ◦ Xu −τ
τ + ∇p
τ = ∆uτ, (5)
for some pτ ∈ C∞ with R pτdLd= 0.
Proof. Existence follows by standard weak compactness-lower semicontinuity arguments. For uniqueness, observe that the map v 7→R |∇v|2 is convex and
v 7→ Z |v ◦ Xτu− u|2dLd= Z |v − u ◦ X−τu |2dLd, is strictly convex.
To write the Euler equation, notice that for ξ ∈ C∞(Td, Rd) with ∇ · ξ = 0, the standard perturbation argument gives
Z uτ − u ◦ Xu −τ τ · ξdL d= − Z ∇uτ∇ξdLd,
thus uτ, pτ is a weak solution of the Stokes problem (5). Standard regularity theory for the
Now we use this minimization problem to build a time-discretized solution of the Navier-Stokes equation:
Definition 3.2 (Discrete solutions) Let u ∈ C∞(Td, Rd) be a smooth vector field with ∇·u = 0 and τ > 0. Define the vector field (uτt) recursively by:
uτ0 := u, uτ(n+1)τ := argmin ∇·v=0 F (v; uτnτ, τ ), ∀n ∈ N uτt := uττ [t τ] , ∀t ≥ 0.
The discrete pressure field (pτt) is defined, for every t ≥ 0, by
∆pτt = ∇ · u τ t ◦ X uτ t −τ τ ! , Z pτtdLd= 0.
Notice that since the smoothness of a vector field implies the smoothness of the corresponding minimizer for any τ > 0, the flow maps are always well defined and so are the discrete solutions. Also uτt, pτt are smooth for any t ≥ 0.
4
The results
4.1 Weak solutionsHere we prove that discrete solutions produce, when τ ↓ 0, weak solutions of the Navier-Stokes equations in any dimension.
Definition 4.1 (Hopf Solutions) We say that (ut) is a Hopf solution of the Navier-Stokes equation starting from u provided it satisfies
Z Z (0,∞)×Td − hut, ∂tξti − hut, ∇ξt· uti + h∇ut, ∇ξti dt dLd= Z Td hu, ξ0i dLd, (6)
for any ξ ∈ Cc∞([0, ∞) × Td, Rd) with ∇ · ξt= 0 for any t ≥ 0, 1 2kusk 2 L2 + Z s 0 k∇urk2L2dr ≤ 1 2kuk 2 L2, ∀s ≥ 0, 1 2kusk 2 L2 + Z s t k∇urk2L2dr ≤ 1 2kutk 2 L2, a.e. t > 0, ∀s ≥ t,
Proposition 4.2 (One step estimates) With the same notation and assumptions of Propo-sition 3.1 it holds:
• Discrete energy inequality.
1 2ku τk2 L2+ τ k∇uτk2L2 ≤ 1 2kuk 2 L2. (7)
• Discrete distributional solution. For every ξ ∈ C∞(Td, Rd) it holds uτ− u τ , ξ L2 − u,ξ ◦ X u τ − ξ τ L2
+ hpτ, ∇ · ξiL2 = huτ, ∆ξiL2 = − h∇uτ, ∇ξiL2 (8)
• Discrete uniform weak continuity. For any ξ ∈ C∞(Td, Rd) with ∇ · ξ = 0 it holds | huτ− u, ξiL2| ≤ τ C(ξ) kukL2 + kuk2L2 , (9) where C(ξ) := max{Lip(ξ), k∆ξkL2}
• Rough estimate on the discrete time derivative.
uτ − u τ HDf−m ≤ C(kukL2 + kuk2L2) (10)
for any m > n2+ 2, for some constant C independent on u, τ , where HDf−m is the dual space of the space of divergence free vector fields in Hm(Td, Rd) with 0 mean, endowed with the norm kuk2
Hm :=
P
αk∂αuk2L2, where α varies over all the multiindexes of length m.
Proof. To get (7) multiply (5) by τ uτ and integrate to get
kuτk2L2 −uτ, u ◦ X−τu
L2 = τ hu
τ, ∆uτi L2,
and conclude noticing that
huτ, ∆uτiL2 = −k∇uτk2L2, uτ, u ◦ Xu −τ L2 ≤ 1 2ku τk2 L2 + 1 2ku ◦ X u −τk2L2 = 1 2ku τk2 L2 + 1 2kuk 2 L2.
To get (8) we sum and subtract u/τ , multiply (5) by ξ, and integrate by parts the terms involving the pressure and ∆uτ. For the discrete convective term we use the fact that (Xtu)#Ld= Ld for any t and thus, by the semigroup property of (Xtu),
u ◦ Xu −τ, ξ L2 = hu, ξ ◦ X u τiL2.
For (9) we observe that Z ξ ◦ Xτu− ξ 2 dLd= Z Z τ 0 ∂t(ξ ◦ Xtu)dt 2 dLd = Z Z τ 0 ∇ξ ◦ Xu t · u ◦ Xtudt 2 dLd ≤ τ Z Z τ 0 |∇ξ ◦ Xtu· u ◦ Xtu|2dtdLd = τ Z τ 0 Z |∇ξ ◦ Xtu· u ◦ Xtu|2dLddt = τ Z τ 0 Z |∇ξ · u|2dLddt ≤ τ2Lip2(ξ)kuk2L2, (11) yields kξ ◦ Xτu− ξkL2 ≤ τ Lip(ξ)kukL2, (12)
and conclude using (8).
It remains to prove (10). Start recalling that for any ξ ∈ Hm(Td, Rd), it holds k∆ξkL2 + Lip(ξ) ≤ C kξkHm,
for some constant C. Therefore from (9) we get
| huτ− u, ξiL2| ≤ Cτ kξkHm kukL2+ kuk2L2 , ∀ξ ∈ C∞(Td, Rd) s.t. ∇ · ξ = 0, which implies (10).
Theorem 4.3 (Hopf solutions) For any sequence τn ↓ 0 there exists a subsequence, not re-labeled, such that uτn
t weakly converges (in L2(Td)) to some ut as n → ∞ for any t ≥ 0, the convergence is strong for a.e. t and (uτn
t ) converges strongly in L2loc([0, +∞), L2(Td, Rd)) to (ut). Any limit vector field (ut) found in this way is a Hopf solution of the Navier–Stokes equations. Proof.
Compactness. From (7) we immediately get
kuτtkL2 ≤ kukL2, ∀t, τ > 0. (13)
Thus with a diagonalization argument, for each sequence τn ↓ 0 we can find a subsequence, not relabeled, such that for each rational t, the sequence n 7→ uτn
t weakly converges to some ut ∈ L2(Td, Rd). From (9) we easily get that for every ξ ∈ C∞(Td, Rd) such that ∇ · ξ = 0 it holds,
huτt − uτs, ξiL2
which is enough to conclude that there is weak convergence for every t ≥ 0 as τn↓ 0.
To get the strong convergence we use the Aubin-Lions lemma. In order to do so it is better to introduce the piecewise affine interpolation of the {uτnτ}n∈Nin place of the piecewise constant one: wtτ := 1 − t τ + t τ uττ [t τ] + t τ − t τ uττ ([t τ]+1) . (15)
It is immediate to verify that the compactness of {(wτ
t)}τ implies that of {(uτt)}τ. To get the compactness of {(wτt)}τ in L2loc([0, ∞), L2) we apply the Aubin-Lions lemma to the spaces
X :=nu ∈ H1(Td, Rd) : ∇ · u = 0, Z Td u Ld= 0o, Y :=nu ∈ L2(Td, Rd) : ∇ · u = 0, Z Td u Ld= 0o, Z := HDf−m,
where X, Y are endowed with the H1 and the L2 norms respectively. Then from (10) and the definition of (wτt) we get that
k∂twτtkH−m Df = uτt+τ − uτ t τ HDf−m
≤ C(kukL2+ kuk2L2), a.e. t,
which is sufficient to conclude.
Now that we have compactness in L2loc([0, +∞), L2(Td, Rd)), we know that for any sequence τn↓ 0 there exists a subsequence τnk such that u
τnk
t converges strongly to ut for a.e. t > 0. Any limit is a Hopf solution. Let τn↓ 0 be a sequence for which we have weak convergence for all times and strong convergence for a.e. times and let (ut) be the limit vector field. We have the uniform bound
kutkL2 ≤ lim
n→∞ kuτn
t kL2 ≤ kukL2, ∀t ≥ 0,
and passing to the limit in (14) we obtain
hut− us, ξiL2
≤ (t − s)C(ξ) kukL2 + kuk2L2, ∀t, s ∈ [0, ∞),
which is enough to get the weak continuity of t 7→ ut. Now let A ⊂ [0, ∞) the set of t’s such that uτn
t converges strongly to utas n → ∞ and notice that certainly 0 ∈ A and L1([0, ∞) \ A) = 0. Choose t ∈ A and s > t and observe that from (7) we get 1 2ku τ sk2L2 + τ [sτ]+τ Z τ [τt]+τ k∇uτrk2L2dr ≤ 1 2ku τ tk2L2, ∀τ > 0. (16)
The choice of t ensures that kuτn t kL2 → kutkL2, so that from lim n→∞ 1 2ku τn s k2L2 + τn[sτ n]+τn Z τn[τnt ]+τn k∇uτn r k2L2dr ≥ 1 2kusk 2 L2 + s Z t lim n→∞ k∇uτn r k2L2dr ≥ 1 2kusk 2 L2 + s Z t k∇urk2 L2dr,
we get the energy inequality.
To conclude we need to show that (ut) satisfies (6). Fix ξ ∈ Cc∞([0, ∞) × Td, Rd) such that ∇ · ξt = 0 for every t ≥ 0, and for any k = 0, 1 . . . we consider (8) for u = uτkτ tested with ξkτ. Adding up one gets
− Z Z +∞ τ uτt ·ξτ [ t τ] − ξτ ([t τ]−1) τ + u τ t · ξτ [t τ] ◦ Xuτt τ − ξτ [τt] τ + u τ t · ∆ξτ [τt]dt dLd= Z u · ξ0dLd.
From the smoothness of ξ we know that ξτ [t τ]− ξτ ([ t τ]−1) τ → ∂tξt, ∆ξτ [t τ] → ∆ξt,
uniformly in [0, ∞) × Td as τ ↓ 0. Thus from the strong convergence of (uτn
t ) to (ut) in L2loc([0, ∞), L2) we get Z Z +∞ τn uτn t · ξτn[ t τn]− ξτn([ t τn]−1) τn dt dLd → Z Z +∞ 0 ut· ∂tξtdt dLd, Z Z +∞ τn uτn t · ∆ξτn[τnt ]dt dL d → Z Z +∞ 0 ut· ∆ξtdt dLd= − Z Z +∞ 0 ∇ut· ∇ξtdt dLd,
as n → ∞. Thus to conclude it is sufficient to check that ξτ n[τnt ]◦ X uτnt τ − ξτn[t τn] τn → ∇ξt· ut,
weakly in L2([0, T ], L2) as n → ∞, where T is such that supp(ξ) ⊂ [0, T ] × Td. From (12) and (13) it is easy to deduce that
1 τn ξτn[τnt ]◦ X uτnt τ − ξτn[ t τn] L2([0,∞),L2)≤ T kukL 2sup t Lip(ξt).
Thus to prove the desired weak convergence it is sufficient to prove that for every ˜ξ ∈ Cc∞([0, ∞)× Td, Rd) it holds 1 τn Z Z D ˜ξt, ξ τn[τnt ]◦ X uτnt τn − ξτn[τnt ] E dt dLd → Z D ˜ξt, (ut· ∇)ξtE dt dLd, as n → ∞.
To prove this, recall that ξτn[ t τn] ◦ Xuτnt τn − ξτn[τnt ] τn = 1 τn Z τn 0 (uτn t · ∇)ξτn[τnt ] ◦ Xuτnt s ds, and therefore 1 τn Z Z D ˜ξt, ξ τn[τnt ]◦ X uτnt τn − ξτn[ t τn] E dt dLd = 1 τn Z Z Z τn 0 D ˜ξt, (uτn t · ∇)ξτn[τnt ] ◦ Xuτnt s E ds dt dLd = 1 τn Z τn 0 Z Z D ˜ξt, (uτn t · ∇)ξτn[τnt ] ◦ Xuτnt s E dLddt ds = 1 τn Z τn 0 Z Z D ˜ξt◦ Xuτnt −s , (uτtn· ∇)ξτn[τnt ] E dLddt ds = Z Z * 1 τn τn Z 0 ˜ ξt◦ Xu τn t −s ds, (uτtn· ∇)ξτn[τnt ] + dLddt. (17)
To conclude notice that the strong convergence of (uτn
t ) to (ut) and the smoothness of ξ ensure that (uτn
t · ∇)ξτn[τnt ]strongly converges to (ut· ∇)ξtin L
2([0, ∞), L2). Also, from (12) with τ = s one gets Z Z 1 τn Z τn 0 ˜ ξt◦ X uτnt −s ds − ˜ξt 2 dLddt = 1 τn Z τn 0 Z ˜ ξt− ˜ξt◦ X uτnt s 2 L2dtds ≤ 1 τn Z τn 0 Z Te 0 s2Lip( ˜ξ)2kuτn t k 2 L2dtds ≤ τ 2 n 3 kuk 2 L2T Lip( ˜e ξ)2, (18)
where eT is such that supp( ˜ξ) ⊆ [0, eT ] × Td. Therefore τ1
n τn R 0 ˜ ξt◦X uτnt −s ds − ˜ξt L2([0,∞),L2)= O(τn),
which completes the proof.
4.2 Suitable solutions in dimension 3
Here we show that in dimension 3 the discrete solutions converge to suitable solutions of the Navier-Stokes equations.
Definition 4.4 (Suitable solution) We say that the pair (ut, pt) is a suitable solution of the Navier-Stokes equations starting from u provided it is a distributional solution of the
Navier-Stokes equation, (ut) is a Hopf solution starting from u, and in addition (ut) ∈ L3
loc([0, ∞), L3(T3, R3)), (pt) ∈ L3/2loc([0, ∞), L3/2(T3)) and it holds ∂t |ut|2 2 + ∇ · ut |ut|2 2 + pt + |∇ut|2≤ ∆ |ut|2 2 , (19)
in the sense of distributions, that is
Z Z −|ut| 2 2 ∂tϕ − ∇ϕ · ut |ut|2 2 + pt + ϕ|∇ut|2dt dL3≤ Z Z ∆ϕ|ut| 2 2 dt dL 3,
for any non negative ϕ ∈ Cc∞((0, ∞) × T3).
We recall that the importance of suitable solutions is due to the work [2] where important partial regularity results for these solutions have been achieved.
Lemma 4.5 (A time step) With the same notation and assumptions of Proposition 3.1 it holds Z 1 2(|u τ|2− |u|2)ϕ − 1 2|u| 2(ϕ ◦ Xu τ − ϕ) − ∇ϕ · uτpτ+ |∇uτ|2ϕdLd≤ τ Z |uτ|2 2 ∆ϕdL d,
for any nonnegative ϕ ∈ C∞(Td).
Proof. Multiply (5) by uτ, by ϕ ∈ C∞(Td) with ϕ ≥ 0 and integrate to get Z |uτ|2ϕ − uτ· u ◦ X−τu ϕ + ϕuτ · ∇pτdLd= τ Z uτ· ∆uτϕdLd. It holds Z uτ · ∆uτϕdLd= − Z |∇uτ|2ϕ −X i,j ui∂jui∂jϕ dLd = − Z |∇uτ|2ϕdLd+ Z |uτ|2 2 ∆ϕdL d, Z ϕuτ· ∇pτdLd= − Z ∇ϕ · uτpτdLd, Z uτ· u ◦ X−τu ϕdLd≤ 1 2 Z |uτ|2ϕdLd+ 1 2 Z |u ◦ X−τu |2ϕdLd = 1 2 Z |uτ|2ϕdLd+ 1 2 Z |u|2ϕ ◦ XτudLd,
Lemma 4.6 (Estimate on the pressure) For any divergence free vector field u ∈ L3(Td, Rd) ∩ C∞(Td, Rd) and τ > 0 define pτu as the only solution of
∆pτu = ∇ · u ◦ X u −τ τ , R pτ udLd= 0. (20)
Then for some C > 0 independent on u it holds
kpτukL3/2≤ Ckuk2L3. (21)
Proof. For any ϕ ∈ C∞(Td) we have Z pτu∆ϕ dLd= Z u ◦ Xu −τ − u τ · ∇ϕ dL d = Z u ·∇ϕ ◦ X u τ − ∇ϕ τ dL d = Z u · 1 τ Z τ 0 ∇2ϕ · u ◦ Xu s ds dLd ≤ kukL3 1 τ Z τ 0 ∇2ϕ · u ◦ Xu s ds L3/2 ≤ kukL3 1 τ Z τ 0 ∇2ϕ · u ◦ Xsu L3/2 ds = kukL3 ∇2ϕ · u L3/2 ≤ kuk2L3 ∇2ϕ L3. (22)
Standard elliptic regularity results ensure that k∇2ϕkL3 ≤ Ck∆ϕkL3 for some C > 0. Thus we
get Z pτu∆ϕdLd ≤ Ckuk2L3k∆ϕkL3.
Again by standard arguments, we know that the equation ∆ϕ = ψ has a smooth solution for every smooth ψ such thatR ψdLd= 0. Thus the above bound yields
Z pτuψ dLd
≤ Ckuk2L3kψkL3, ∀ψ ∈ C∞(Td) such that
Z
ψ dLd= 0.
Since R pτudLd= 0, from the last inequality we conclude that (21) is true. Theorem 4.7 (Suitable solutions in dimension 3) Let (uτ
t) and (pτt) be defined by 3.2. Then {(uτt)}τ is compact in L3loc([0, ∞), L3(T3, R3)) and {(pτt)}τ is weakly compact in L3/2loc([0, ∞), L3/2(T3)). For any sequence τn ↓ 0 such that (uτtn) strongly converges to some (ut) in L3loc([0, ∞), L3(T3, R3)) and (p τn t ) weakly converges in L 3/2 loc([0, ∞), L 3/2(T3)) to some
Proof.
Compactness. Arguing as in the proof of Theorem 4.3 and using the compact embedding of H1 into L4, we deduce that {(uτt)}τ is relatively compact in L2loc([0, ∞), L4). For any (ut) ∈ L2loc [0, ∞), L4 ∩ L∞ [0, ∞), L2 it holds
Z Z
|ut|3dL3dt ≤ Z
kutkL2kutk2L4dt = kukL∞([0,∞),L2)kuk2L2([0,+∞),L4),
and therefore the uniform bound (13) ensures the desired strong relative compactness of {(uτ t)}τ in L3([0, +∞), L3). To get the weak compactness of {(pτt)}τ, notice that pτt is the unique solution of (20) for u = uτt, and from the uniform bound of (uτt) in L3([0, +∞), L3), together with (21), we get weak compactness of (pτ
t) in L3/2([0, +∞), L3/2).
Limits are suitable solutions. To simplify the notation, we assume that (uτt) → (ut) strongly in L3([0, ∞), L3) (pτt) → (pt) weakly in L3/2([0, ∞), L3/2) (i.e., we are not considering a sequence τn ↓ 0). Theorem 4.3 guarantees that u is a Hopf solution of the Navier-Stokes system, and clearly (ut) ∈ L3([0, ∞), L3), (pt) ∈ L3/2([0, ∞), L3/2).
To prove that (ut), (pt) is a distributional solution, fix ξ ∈ Cc∞((0, ∞) × T3, R3), and for any k = 0, 1 . . . consider (8) for u = uτ
kτ, tested with ξkτ. Adding up one gets
Z Z +∞ τ uτt·ξτ [ t τ]− ξτ ([ t τ]−1) τ +u τ t· ξτ [t τ]◦ X uτ t τ − ξτ [τt] τ dt dL 3 = Z Z +∞ τ uτt·∆ξτ [t τ]+p τ t∇·ξτ [τt]dt dL3. Now since ∇ · ξτ [t τ] → ∇ · ξt uniformly on [0, +∞) × T
3, the weak convergence of (pτ
t) to (pt) yields Z Z pτt∇ · ξτ [t τ]dt dL 3 → Z Z pt∇ · ξtdt dL3,
while all the other terms are treated as in the proof of theorem 4.3.
To prove the generalized energy inequality, fix a non negative ϕ ∈ Cc∞((0, ∞) × T3). We want to show that
Z Z −|ut| 2 2 ∂tϕ + ut· ∇ϕ |ut|2 2 + pt + |∇ut|2ϕdtdL3 ≤ Z Z |u t|2 2 ∆ϕdt dL 3
Suppose τ is sufficiently small, such that supp(ϕ) ⊂ [τ, +∞). From Lemma 4.5 and the definition of uτ, pτ we immediately get, with the usual argument
Z Z −|u τ t|2 2 ϕτ [t τ] − ϕτ ([t τ]−1) τ − |uτ t|2 2 ϕτ [t τ] ◦ Xuτt τ − ϕτ [t τ] τ − ∇ϕτ [τt] · uτtpτt + |∇uτt|2ϕτ [t τ]dtdL 3 ≤ Z Z |uτ t|2 2 ∆ϕτ [τt]dtdL 3.
The convergence of uτ ensures that |uτ|2 converges to |u|2 in L3/2([0, ∞), L3/2), and from the smoothness of ϕ and the weak convergence of pτ it is immediate to verify that as τ ↓ 0 it holds
Z Z |uτ t|2 2 ϕτ [t τ]− ϕτ ([ t τ]−1) τ dtdL 3 → Z Z |u t|2 2 ∂tϕ dtdL 3, Z Z ∇ϕτ [t τ]· u τ tpτtdtdL3 → Z Z ∇ϕt· utptdtdL3, Z Z |uτ t|2 2 ∆ϕτ [τt]dtdL 3 → Z Z |u t|2 2 ∆ϕtdtdL 3.
Also, the non negativity of ϕ easily implies
lim inf τ ↓0 Z Z |∇uτt|2ϕτ [t τ]dtdL 3 ≥ Z Z |∇ut|2ϕtdtdL3.
Thus to conclude it is sufficient to show that
Z Z |uτ t|2 2 ϕτ [t τ]◦ X uτ t τ − ϕτ [t τ] τ dtdL 3 → Z Z |uτ t|2 2 ∇ϕt· utdtdL 3, (23) as τ ↓ 0. Since |uτt|2→ |u
t|2 in L3/2([0, ∞), L3/2), to prove (23) it is sufficient to check that
ϕτ [t τ] ◦ Xuτt τ − ϕτ [t τ] τ → ∇ϕt· ut weakly in L 3([0, ∞), L3). From ϕτ [t τ] ◦ Xuτt τ − ϕτ [t τ] τ = 1 τ Z τ 0 ∇ϕτ [t τ]◦ X uτ t s · uτt ◦ X uτ t s ds, we immediately get that it holds
ϕτ [t τ] ◦ Xuτt τ − ϕτ [τt] τ L3([0,∞),L3) ≤ sup t Lip(ϕt)kukL3([0,∞),L3),
which gives weak convergence for some subsequence. To conclude fix ψ ∈ Cc∞((0, ∞) × T3) and notice that Z Z ψt ϕτ [t τ]◦ X uτ t τ − ϕτ [t τ] τ dtdL 3= 1 τ Z Z Z τ 0 ψt∇ϕτ [t τ]◦ X uτt s · uτt ◦ X uτt s dsdtdL3 = 1 τ Z τ 0 Z Z ψt∇ϕτ [τt]◦ X uτ t s · uτt ◦ X uτ t s dL3dtds = 1 τ Z τ 0 Z Z ψt◦ Xu τ t −s∇ϕτ [t τ]· u τ tdL3dtds = Z Z 1Z τ ψt◦ X uτ tds ∇ϕ t · uτdL3dt.
Since ∇ϕτ [t τ]
· uτ
t → ∇ϕt· ut strongly in L3([0, ∞), L3), and, as in (18), we get
1 τ Z τ 0 ψt◦ X uτ t −sds → ψt, in L2([0, +∞), L2), (23) is proved.
Remark 4.8 We remark that one can actually prove that if τn ↓ 0 is such that (uτtn) strongly converges to some (ut) in L3loc([0, ∞), L3), then (p
τn
t ) weakly converges in L 3/2
loc([0, ∞), L3/2) to the distributional solution p of
(
∆pt= ∇ · (ut· ∇)ut, R ptdL3 = 0,
and thus to the pressure term of the limit equation. Indeed, as in the proof of lemma 4.6, it suffices to check that, for a.e. t, one has
Z pτn
t ∆ϕ dL3 →
Z
pt∆ϕ dL3, ∀ϕ ∈ C∞(T3), as n → ∞. To prove this, start from
Z pτn t ∆ϕ dL 3=Z uτn t · 1 τn Z τn 0 ∇2ϕ · uτn t ◦ X uτnt s ds dL3,
(which follows as in (22)), and notice that from the strong convergence of uτn
t to ut in L3 it is sufficient to check that it holds
1 τn Z τn 0 ∇2ϕ · uτn t ◦ X uτnt s ds → ∇2ϕ · ut,
weakly in L3/2. We already know that the norms are uniformly bounded, thus the conclusion follows along the same lines of the last part of the proof of Theorem 4.3 (see in particular (17) and the conclusion thereafter – in the current situation there is no integral in t), we omit the
details.
4.3 Strong solutions
Here we show how from the discretization scheme discussed, one can prove two classical re-sults concerning smooth solutions of the Navier-Stokes equations: existence for a time of order k∇uk−4L2, and existence for all times if u satisfies the classical smallness condition.
Notice that the calculations that we do here are classical: what we want to show is that the standard approach has a natural ‘discrete analogous’ in our setting. We recall that the two estimates (3) hold if we seek for solutions ut of (1) such thatR utdL3≡ 0 for any t ≥ 0, and we can certainly do so, by the discussion preceding the latter formula.
Lemma 4.9 (One step estimates) Let d = 3. With the same notation and assumptions of Proposition 3.1 it holds 1 2k∇u τk2 L2 − 1 2k∇uk 2 L2 + τ 2k∆u τk2 L2 ≤ Cτ k∇uk3L2k∆ukL2 (24) and 1 2k∇u τk2 L2− 1 2k∇uk 2 L2+ τ 2k∆u τk2 L2 ≤ Cτ k∇uk2L2k∆uk2L2. (25)
Proof. Multiplying (5) by −∆uτ and integrating we get
− Z
uτ· ∆uτ+ u · ∆uτ+ (u ◦ X−τu − u) · ∆uτdL3= −τ k∆uτk2L2,
hence after integration by parts and by Young inequality we have 1 2k∇u τk2 L2− 1 2k∇uk 2 L2 + τ k∆uτk2L2 ≤ k∆uτkL2ku ◦ X−τu − ukL2 ≤ τ 2k∆u τk2 L2 + 1 2τku ◦ X u −τ − uk2L2. (26)
With the same calculations we did for (11) we get
ku ◦ Xu
−τ − uk2L2 ≤ τ2
Z
|∇u|2|u|2dL3. (27)
To get (24) we bound the right hand side as
Z
|∇u|2|u|2dL3≤ k∇uk
L6k∇ukL2kuk2L6 ≤ Ck∆ukL2k∇uk3L2,
and plugging this into (26) we get (24)
To get (25), we bound the right hand side of (27) as Z
|∇u|2|u|2dL3 ≤ k∇uk2L6kukL6kukL2 ≤ Ck∆uk2L2k∇uk2L2,
which gives the statement when inserted into (26).
Proposition 4.10 (Strong Solutions) With the same notation and assumptions of Proposi-ton 3.1 and Definition 3.2, there exists a constant C > 0 such that for T := k∇ukC 4 the discrete
solutions converge to a strong solution of the Navier-Stokes equations in [0, T ).
Also, there is another constant C0 such that if k∇ukL2 ≤ C0, then the scheme converges to
the smooth solution of the Navier-Stokes equations in the whole [0, ∞).
Proof. It is well known (see e.g. [4], Theoreom 5.2 for the case of bounded, smooth open subsets of R3) that if a weak solution (ut) of Navier-Stokes belongs to
then it is smooth on [0, T ]. Hence to prove the statement it is sufficient to check that for any ε > 0, the discrete solutions are uniformly bounded both in L∞([0, T ], H1) and in L2([0, T ], H2). Let us fix τ for the moment, and consider (24) and (25) for u = uτiτ, for some nonnegative integer i. Adding up the inequalities (24) from i = 0 . . . n − 1 we get
k∇uτnτk + τ n X i=1
k∆uτiτk2L2 ≤ k∇uk2L2+ 2Cτ
n−1 X i=0
k∇uτiτk3L2k∆uτiτkL2
= k∇uk2L2+ 2C Z nτ 0 k∇uτtk3L2k∆uτtkL2dt ≤ k∇uk2L2+ 2C sup t<nτ k∇uτtk3L2 Z nτ 0 k∆uτtkL2dt ≤ k∇uk2L2+ 2C sup t<nτ k∇uτ tk3L2 √ nτ Z nτ 0 k∆uτ tk2L2dt 12 , and therefore sup t<(n+1)τ k∇uτtk2 ! + Z (n+1)τ 0 k∆uτtk2L2dt
≤ k∇uk2L2 + τ k∆uk2L2 + 2C sup
t<nτ k∇uτtk3L2 √ nτ Z nτ 0 k∆uτtk2L2dt 1 2 . (28)
We can proceed in a similar manner for (25), obtaining
sup t<(n+1)τ k∇uτtk2 ! + Z (n+1)τ 0 k∆uτtk2L2dt
≤ k∇uk2L2+ τ k∆uk2L2 + 2C sup
t<nτ k∇uτtk2L2 Z nτ 0 k∆uτtk2L2dt. (29)
Let us fix T > 0 and suppose 0 ≤ n ≤Tτ. We define δτ(u) := k∇uk2L2+ τ k∆uk2L2, an:= sup
t<nτ
k∇uτtk2+ Z nτ
0
k∆uτtk2L2dt,
and notice that Young inequality applied to the last term on the right of both (28) and (29), yields
an+1≤ δτ(u) + C √
T a2n, an+1≤ δτ(u) + Ca2n, respectively, and thus
an+1≤ δτ(u) + C min{1, √
T }a2n. Now, suppose that the equation λ = δτ(u) + C min{1,
√
T }λ2 has a positive solution, i.e.,
min{1,√T }δτ(u) ≤ 1
and let λ be the smallest one: λ = 2δτ(u) 1 + q 1 − 4C min{1,√T }δτ(u) ≤ 2δτ(u).
It is easily proved by induction that an ≤ λ for any n, since a0 ≤ δτ(u) ≤ λ. Therefore the family {uτt} is bounded both in L∞([0, T − ε], H1) and in L2([0, T − ε], H2) by 2δτ(u). For τ ≤ k∇uk2L2/ k∆uk2L2 we get δτ(u) ≤ 2 k∇uk2L2, and condition (30) reads
min{1, √
T } k∇uk2L2 ≤
1 8C,
which gives the claims.
5
The case of bounded and smooth open sets
All the conclusions of this paper are still true if we work on a bounded smooth open subset Ω of R3 (weak solutions are produced in any dimension) provided we add the Dirichlet boundary condition ut ≡ 0 on ∂Ω. In this case the minimization problem of proposition 3.1 has to be solved in the set H1(Ω) := {v ∈ H01(Ω) : ∇ · v = 0}, and all the results follow along the same line. The only difference from the periodic case is in the proof of the L3/2loc([0, +∞), L3/2(Ω)) bound on the pressure pτ, needed in passing to the limit when one wants to prove suitability of the solution. To this end one has to proceed in a different way, using the mixed Lp,sestimates of the evolutionary Stokes problem. The space Ls,r([0, ∞) × Ω)) := Ls([0, ∞); Lr(Ω)) is equipped with the norm
kvkLs,r := Z ∞ 0 kvtksLrdt 1/s Lemma 5.1 Let uτ
t be a discrete solution of the Navier–Stokes equation, as in definition 3.2, for some smooth initial data u with ∇ · u = 0 and u|∂Ω= 0. Then
uτ − uτ ◦ Xu τ −τ Ls,r ≤ τ C(u) (31)
for 3/2 ≥ r ≥ 1 and s ≥ 1 such that
4 = 3 r +
2
s (32)
Proof. Denoting by r0 the conjugate exponent of r, we follow the proof of (11) with exponent r instead of 2, obtaining, for any smooth u
Z Ω |u − u ◦ X−τu |rdL3= Z Ω Z τ 0 d dtu ◦ X u −tdt r dL3 ≤ τr/r0 Z τ 0 Z Ω |∇u ◦ X−tu · u ◦ X−tu |rdL3dt ≤ τr Z |∇u · u|rdL3.
By Holder inequality with exponent 2/r on ∇u, and by the interpolation inequality
kukLq(Ω) ≤ C k∇uk1−αL2(Ω)kuk
α L2(Ω), 2 ≤ q ≤ 6, α = 3 q − 1 2, for q = 2r/(2 − r), one gets
k∇u · ukLr(Ω) ≤ kuk L2−r2r (Ω)k∇ukL2(Ω) ≤ Ckuk 3 r−2 L2(Ω)k∇uk 4−3r L2(Ω).
We apply this estimate for u = uτt, integrate in time over [0, ∞), use (16) and (32) to get
uτ− uτ◦ X−τuτ s Ls,r ≤ Cτ Z ∞ 0 kuτtk2(s−1)L2(Ω)k∇u τ tk2L2(Ω)dt ≤ Cτ kuk2(s−1)L2(Ω) Z ∞ τ k∇uτtk2L2(Ω)dt + τ k∇uk2L2(Ω) ≤ Cτ kuk2sL2(Ω)+ τ kuk 2(s−1) L2(Ω)k∇uk2L2(Ω).
Finally, the condition 2 ≤ 2r/(2 − r) ≤ 6 forces 1 ≤ r ≤ 3/2.
To obtain a bound for the pressure pτ one can then proceed as in [8]: one lets
gnτ = P u τ n− uτn◦ X uτ n −τ τ ! t=nτ
where P is the projection on the closure in Lr(Ω) of {u ∈ Cc∞: ∇ · u = 0} (the dependance on r will be implicit) and uτn= uτ(nτ ). Then uτn solves the difference equation in P Lr(Ω)
(
uτn+1− uτ
n= τ Suτn+1+ τ gτn,
uτ0 = u, (33)
where S = P ∆ is the Stokes operator in P Lr(Ω). We recall here that, (see [9], [5]), S generates an analytic semigroup with optimal regularity in P Lr(Ω). Setting Σθ:= {λ ∈ C \ {0} : |Arg(λ)| ≤ θ}, this is equivalent, by e.g. [6, Theorem 1.11] to
{λR(λ, S) : λ ∈ Σθ} R-bounded for some θ ≥ π/2,
(and thus bounded). We will henceforth fix such a θ ≥ π/2 and let R(S) be the corresponding R-bound.
We rewrite the difference equation in the standard form ( uτn+1= Tτuτn+ fnτ, uτ0 = u, (34) where Tτ = (1 − τ S)−1, fnτ = τ (1 − τ S) −1gτ n.
Now, Tτ is a powerbounded operator in P Lr(Ω), since σ(Tτ) = 1 1 + τ λn : λn∈ σ(S) ⊂ (−1, 0),
and it is also analytic, i.e.
{n(Tτ− 1)Tτn} is bounded. (35)
To prove analyticity is suffice to observe that
Tτ− 1 = τ S(1 − τ S)−1 := τ Sτ (36)
is (a multiple of) the Yoshida approximation Sτ of S in P Lr(Ω), and thus it generates an analytic semigroup; this, together with σ(Tτ) ⊂ {z ∈ C : |z| ≤ 1} is equivalent to (35) by [1, Theorem 2.3]. We can reduce system (34) to zero initial data by subtracting
vnτ = Tτnu, noticing that vn+1τ − vτ n Lr(Ω)≤ 1 nkn(Tτ− 1)T n τk kukLr(Ω). (37)
We therefore seek for (uniform in τ ) optimal Ls N, P Lr(Ω)-regularity of the discrete time parabolic equation for wn:= un− vn
(
wτn+1= Tτwnτ+ fnτ, wτ0 = 0,
in the sense that
+∞ X n=1 wn+1τ − wτn s Lr(Ω) ≤ Cs,r ∞ X n=1 kfnτksLr(Ω). (38)
By [1, Theorem 1.1], the discrete optimal Ls N, P Lr(Ω)-regularity is equivalent to the optimal Ls [0, +∞), Lr(Ω) regularity of the analytic semigroup generated by the operator Sτ given in (36). Moreover, the constant in the optimal regularity estimate depends only on the R-bound of
{itR(it, Sτ) : t ∈ R \ {0}}, (39)
and thus it suffice to prove an R-bound for this set of operators which is indipendent of τ . To this end, a short computation shows that
itR(it, Sτ) = 1 itλ 2 τ(t)R(λτ(t), S) + τ λτ(t)Id where λτ(t) = it .
Notice that λ(R\{0}) is the circle through the origin, with center on the real axis and radius 1/τ , minus the origin; therefore it is contained in Σπ/2. By the subadditivity and submultiplicativity of the R-bounds, we get that the set in (39) is R-bounded by R(S) + 1, since {λτ(t)R(λτ(t), S) : t ∈ R \ {0}} is R-bounded by R(S) and
|λτ(t)/t| ≤ 1, τ |λτ(t)| ≤ 1, ∀t ∈ R.
This shows that the constant in (38) (and thus, a fortiori, the one in (35)) is bounded indipen-dently of τ . We are now ready to prove an estimate of the pressure, which allows to prove the suitability of the limit solutions, in the case of Ω bounded and smooth.
Theorem 5.2 Let Ω ⊂ R3 be a smooth and bounded open set. Let uτt, pτt be a discrete solution of the Navier–Stokes equation, as in definition 3.2, for some smooth initial data u with ∇ · u = 0 and u|∂Ω= 0. For any ε > 0, there exists a constant C = C(ε, u) such that for any sufficiently small τ > 0
kpτkL5/3(Ω×[ε,∞)) ≤ C(ε, u).
Proof. Let r and s satisfy (32), with s > 1, and let m = [ε/τ ]. With the same notation of the preceding discussion we first of all prove that for some constant C independent of τ , it holds
+∞ X n=m uτn+1− uτ n τ s Lr(Ω) ≤ C +∞ X n=0 kgnτksLr(Ω)+ C εs−1τ kuk s Lr(Ω). (40)
By (37), for any n ≥ m it holds
vτn+1− vτ n τ Lr(Ω) ≤ C 1 nτ kukLr(Ω), and thus +∞ X n=m vτn+1− vτ n τ s Lr(Ω) ≤ C +∞ X n=m 1 nsτskuk s Lr(Ω) ≤ C τ 1 (τ m)s−1 kuk s Lr(Ω) ≤ C εs−1τ kuk s Lr(Ω) (41)
Moreover, by the Lr(Ω) resolvent estimate for the Stokes operator, it holds (1 − τ S)−1 ≤ C, and thus
kfτ
nkLr(Ω) ≤ Cτ kgτnkLr(Ω).
We split uτn= wnτ+ vτn and use this estimate and (38) for wn and (41) for vn, obtaining (40). Notice that the difference equation (33) readily gives an analogous estimate for Suτn+1
Lr(Ω);
finally we can take the Lr(Ω) norm in the original difference equation
∇pτn= ∆uτn−u τ n+1− uτn τ + uτn− uτ n◦ X uτ n −τ τ ,
and by the coercive estimate kukW2,r(Ω) ≤ C kSukLr(Ω), obtain the full regularity estimate +∞ X n=m uτn+1− uτ n τ s Lr(Ω) + ∆uτn+1 s Lr(Ω)+ ∇pτn+1 s Lr(Ω) ≤ C +∞ X n=0 uτ n− uτn◦ X uτ n −τ τ s Lr(Ω) +Cε τ kuk s Lr(Ω),
where Cε= C/εs−1. We now choose s = 5/3 and r = 15/14, and use lemma 5.1 in this estimate to get k∇pτk5/3 L5/3([ε,∞);L15/14(Ω))≤ τ +∞ X n=m ∇pτn+1 5/3 L15/14(Ω) ≤ C(ε, u). SinceR
ΩpτdL3 = 0, the Sobolev-Poincar´e inequality kpkL5/3(Ω) ≤ C k∇pkL15/14(Ω)applied to the
latter estimate gives the claim.
Remark 5.3 (On the smoothness of the initial datum) We point out that the restriction to smooth initial data has been made to simplify the discussion about the existence of flow maps. Actually, this is unnecessary: as showed by DiPerna-Lions in [3], as soon as a vector field u has Sobolev regularity and is divergence free, one has existence and uniqueness of a one parameter group of maps Xtu such that (Xtu)#Ld = Ld, X0 = Id and ∂tXtu = u ◦ Xtu in the sense of distributions. Thus one can use these maps as flow maps and directly get a weak/suitable solution for any initial datum in H1. Notice, however, that in order to get the discrete estimates needed for the existence of strong solutions, it is required that the initial datum is in H2, although the H2 norm actually does not appear in the final statement.
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